arXiv:1003.5597v3 [math.GT] 28 Jun 2010
BRIDGEMAN’S ORTHOSPECTRUM IDENTITY
DANNY CALEGARI
Abstract. We give a short derivation of an identity of Bridgeman concerning
orthospectra of hyperbolic surfaces.
1. Introduction
In [1], Martin Bridgeman proves a beautiful identity concerning orthospectra of
hyperbolic surfaces with totally geodesic boundary. Let Σ be a hyperbolic surface
with totally geodesic boundary. An orthogeodesic is a geodesic segment properly
immersed in Σ, which is perpendicular to ∂Σ at its endpoints. The set of orthogeodesics is countable, and their lengths are proper. Denote these lengths by li
(with multiplicity). Bridgeman’s identity is:
Theorem 1.1 (Bridgeman, [1]). With notation as above,
X
L(1/ cosh2 (li /2)) = −π 2 χ(Σ)/2
i
where L is the Rogers’ dilogarithm function (to be defined below).
Treating the function L as a black box for the moment, the identity has the form
X
ℓ(li ) = a term depending only on the topology of Σ
i
The proof is very, very short and elegant. By the Gauss–Bonnet theorem, the term
on the right is equal to 1/8 of the volume of the unit tangent bundle of Σ. Almost
every tangent vector on Σ can be exponentiated to a geodesic on Σ which intersects
the boundary in finite forward and backward time (by ergodicity of the geodesic
flow on a closed hyperbolic surface obtained by doubling; see e.g. [4]). If v is such a
tangent vector, and γv is the associated geodesic arc, then γv is homotopic keeping
endpoints on ∂Σ to a unique orthogeodesic (which is the unique length minimizer
in its relative homotopy class).
The volume of the set of v associated to a given orthogeodesic ω can be computed
as follows. Lift ω to the universal cover, together with lifts of the boundary geodesics
it ends on. The three together make a letter “H” in which ω is the crossbar; see
Figure 1. Any γv in the homotopy class of ω lifts to a unique geodesic segment in
the universal cover with endpoints on the sides of the H. Therefore the volume of
the set of such v depends only on the geometry of the H, which in turn depends
only on l = length(ω). This volume is 8ℓ(l) with notation as above.
To complete the proof of Bridgeman’s identity therefore, it suffices to show ℓ(l) =
L(1/ cosh2 (l/2)). Bridgeman derives this in several pages of calculations. The
purpose of this note is to give a short derivation of this fact, using elementary
hyperbolic geometry.
Date: June 29, 2010.
1
2
DANNY CALEGARI
ω
Figure 1. An orthogeodesic ω lifts to the crossbar of a letter H
2. Derivation of L
The “ordinary” polylogarithms Lik can be defined by a Taylor series
∞
X
zn
Lik (z) =
nk
n=1
which converges for |z| < 1, and extends by analytic continuation. Taking derivatives, one sees that they satisfy the identities
Li′k (z) = Lik−1 (z)/z
thereby giving rising to recursive integral formulae for these functions. The special
case Li0 (z) is the familiar geometric series for z/(1 − z), so Li1 (z) = − log(1 − z)
and
Z
log(1 − z)
dz
Li2 (z) = −
z
The Rogers dilogarithm is a “normalization” of Li2 given by the formula
1
L(z) = Li2 (z) + log(|z|) log(1 − z)
2
for real z < 1. One sees that the Rogers dilogarithm is obtained by symmetrizing
the integrand for the integral expression for Li2 under the involution z → 1 − z:
1 log(1 − z) log(z)
L′ (z) = −
+
2
z
1−z
A classic reference for this material is Lewin [5].
We now explain how to compute the volume of the set of vectors v tangent
to a geodesic γv intersecting the left and right sides of the H associated to an
orthogeodesic ω. The four ideal vertices of the H span an ideal quadrilateral Q.
The diagonals of this quadrilateral subdivide it into four semi-ideal triangles. We
denote the left and right sides of the H as L and R, and the other two edges of
Q as U and D. Similarly, denote the four triangles TL , TR , TU and TD labeled
according to which edge of the quadrilateral they bound; see Figure 2 (the triangle
TR is colored gray in the figure). The crossbar of the H (corresponding to the
orthogeodesic ω itself) is not depicted in the figure.
We identify the ideal circle with RP1 in such a way that the vertices of Q are (in
circular order) 0, x, 1, ∞, where ∞, 0 are the ideal vertices of the gray triangle. Call
α the (hyperbolic) angle of the gray triangle at its vertex. By elementary hyperbolic
trigonometry, x = (1 + cos(α))/2 = tanh2 (l/2) where l is the distance between L
BRIDGEMAN’S ORTHOSPECTRUM IDENTITY
x
3
0
U
L
R
D
1
∞
Figure 2. An H spans an ideal quadrilateral, which is dissected
into four semi-ideal triangles
and R (i.e. the length of the given orthogeodesic). We use the parameters l, x and
α interchangeably in the sequel. We will compute ℓ implicitly as a function of x,
and show that it is a multiple of the Rogers dilogarithm function, thus verifying
Bridgeman’s identity.
Every vector v in Q exponentiates to a (bi-infinite) geodesic γv , and we want to
compute the volume of the set of vectors v for which the corresponding geodesic
intersects both L and R. The point of the decomposition in the figure is that for v
in TL (say), the geodesic γv intersects L whenever it intersects R, so we only need
to compute the volume of the v in TL for which γv intersects R. Similarly, we only
need to compute the volume of the v in TR for which γv intersects L. For v in TU ,
we compute the volume of the v for which γv does not intersect U (since these are
exactly the ones that intersect both L and R), and similarly for TD .
The crux of the matter is that these volumes can be expressed in terms of
integrals of simple harmonic functions. Let χL denote the harmonic function on
the disk which is 1 on the arc of the circle bounded by L, and 0 on the rest of
the circle. This function at each point is equal to 1/2π times the visual angle (i.e.
the length in the unit tangent circle) subtended by the given arc of the circle, as
seen from the given point in the hyperbolic plane. Define χR , χU and χD similarly.
Then the total volume we need to compute is equal to
Z
Z
(1 − 2χU )
4π
2χR +
TU
TL
R
R
(here we have identified TL χR = TR χL by symmetry, and similarly for the other
pair of terms). Let us approach this a bit more systematically. We introduce three
functions A(·), B(·) and C(·) as follows. If α denotes as above the angle at the
nonideal vertex of triangle TR , we define
Z
Z
Z
χL = C(α)
χU = B(α), and
χR = A(α),
TR
TR
TR
The integral we want to evaluate can be expressed easily in terms of explicit
rational multiples of π, and the functions A, B, C. These functions satisfy obvious
identities:
Z
1 − A(α) − 2B(α) = π − α − A(α) − 2B(α)
C(α) =
TR
and
A(α) + B(π − α) = π/3
4
DANNY CALEGARI
where the last identity comes by observing that we are integrating a certain function over an ideal triangle, and observing that the average of this function under
the symmetries of the ideal triangle is equal to the constant function 1/3. In particular, we see that we can express everything in terms of A. After some elementary
reorganization, we see that the contribution V (α) to the volume of the unit tangent
bundle of the surface associated to this particular orthogeodesic is
V (α) = π 2 (8 − 16/3) − 4πα − 8π(A(α) − A(π − α))
It remains to actually compute A(α). To do this it makes sense to move to the
upper half-space model, and move the endpoints of the interval to 0 and ∞. The
harmonic function is equal to 1 on the negative real axis, and 0 on the positive real
axis. It takes the value θ/π on the line arg(z) = θ. The area form in the hyperbolic
metric is proportional to the Euclidean area form, with constant 1/Im(z)2 . In other
words, we want to integrate arg(z)/π Im(z)2 over the region indicated in figure 3,
where the nonideal angle is α, and the base point is 0.
α
0
x
Figure 3.
If we normalize so that the circular arc is part of the semicircle from 0 to 1, then
the real projection of the vertical lines in the figure are 0 and x.
There is no elementary way to evaluate this integral, so instead we evaluate its
derivative as a function of x where as before, x = (1+cos(α))/2. This is the definite
integral
Z ∞
tan−1 (y/x)
dy
A′ (x) =
√
πy 2
y= x−x2
Integrating by parts gives
Z
x
1 ∞
α
+
dy
A′ (x) =
√
2
π sin α π y= x−x2 y(y + x2 )
This evaluates to A′ (x) = (α/π sin α) − 1/π(log(1 − x)/2x). Thinking of V (α) as a
function of x, we get
V ′ (x) = −4πdα/dx − 8π(A′ (x) + A′ (1 − x)) = −8L′ (x)
Comparing values at x = 0 we see that V (x) = 8L(1 − x) (where we use the
symmetry of L′ under x → 1 − x). Substitute x = tanh2 (l/2) and the identity is
proved.
Remark 2.1. The paper [3] by Dupont and Sah relates Rogers dilogarithm to volumes of SL(2, R)-simplices, and discusses some interesting connections to conformal
field theory and lattice model calculations. They cite an older paper of Dupont for
BRIDGEMAN’S ORTHOSPECTRUM IDENTITY
5
the explicit calculations; these are somewhat tedious and unenlightening; however,
he does manage to show that the Rogers dilogarithm is characterized by the Abel
identity. In other words, Dupont shows:
Lemma 2.2 (Dupont [2], Lemma A.1). Let f : (0, 1) → R be a three times differentiable function satisfying
1 − s−1
1 − s1
s2
1
−f
=0
+
f
f (s1 ) − f (s2 ) + f
s1
1 − s2
1 − s−1
2
for all 0 < s2 < s1 < 1. Then there is a real constant κ such that f (x) = κL(x)
where L(x) is the Rogers dilogarithm (up to an additive constant).
Our geometric argument can be reformulated in homological algebraic terms,
though since its virtue is its simplicity, we have not pursued this.
3. Acknowledgments
Danny Calegari was supported by NSF grant DMS 0707130. I would like to
thank the anonymous referee for useful comments, and Martin Bridgeman for a
beautiful talk at Caltech in which he explained his identity. I would also like to
thank Hidetoshi Masai for catching some errors in an earlier version of this paper.
References
[1] M. Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on Moduli
space, arXiv:0903.0683
[2] J. Dupont, The dilogarithm as a characteristic class for flat bundles, J. Pure Appl. Algebra
44 (1987), no. 1-3, 137–164
[3] J. Dupont and C.-H. Sah, Dilogarithm identities in conformal field theory and group homology, Comm. Math. Phys. 161 (1994), no. 2, 265–282
[4] A. Katok and B. Hasselblatt, An introduction to the modern theory of dynamical systems,
Cambridge University Press, Cambridge, 1995
[5] L. Lewin, Dilogarithms and associated functions, Macdonald, London 1958
Department of Mathematics, Caltech, Pasadena CA, 91125